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MATH
[1251]

Theorems from Mass, centroids and probability

All theorems from Mass, centroids and probability.

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Centre of mass and moments

For a lamina with mass density ρ\rho and total mass M>0M>0, the centre of mass is

xˉ=1M∬Ωxρ(x,y) dA,yˉ=1M∬Ωyρ(x,y) dA.{ \bar x=\frac1M\iint_\Omega x\rho(x,y)\,dA, \qquad \bar y=\frac1M\iint_\Omega y\rho(x,y)\,dA }.

The density-weighted first moments are

My=∬Ωxρ(x,y) dA,Mx=∬Ωyρ(x,y) dA.M_y=\iint_\Omega x\rho(x,y)\,dA, \qquad M_x=\iint_\Omega y\rho(x,y)\,dA.

The factor xx weights mass by horizontal position and the factor yy weights it by vertical position. For uniform density, the centre of mass is the centroid, the geometric centre of the region.

Units give a check: if xx and yy are measured in units of length and ρ\rho in mass per unit area, then MM has units of mass, each first moment has units of mass,×\times,length and (xˉ,yˉ)(\bar x,\bar y) has units of length.

  1. [1]:Mass density of a lamina
  2. [2]:A symmetric region with nonconstant density
  3. [3]:Uniform, vertical and radial density

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Centre of mass and moments